Iterative strategies for solving linearized discrete mean field games systems

Abstract

Mean fields games (MFG) describe the asymptotic behavior of stochastic differential games in which thenumber of players tends to $+\infty$. Under suitable assumptions,they lead to a new kind of system of two partial differential equations: a forward Bellman equation coupled with a backward Fokker-Planck equation.In earlier articles, finite difference schemes preserving the structure of the system have been proposed and studied.They lead to large systems of nonlinear equations in finite dimension. A possible way of numerically solving the latter is to use inexact Newton methods: a Newton step consists of solving a linearized discrete MFG system.The forward-backward character of the MFG system makes it impossible to use time marching methods. In the present work, we propose three families of iterative strategies for solving the linearized discrete MFG systems,most of which involve suitable multigrid solvers or preconditioners.